E-characteristic polynomials of real rectangular tensor

Abstract

By the resultant theory, the E-characteristic polynomial of a real rectangular tensor is defined. It is proved that an E-singular value of a real rectangular tensor is always a root of the E-characteristic polynomial. The definition of the regularity of square tensors is generalized to the rectangular tensors, and in the regular case, a root of the Echaracteristic polynomial of a special rectangular tensor is an E-singular value of the rectangular tensor. Moreover, the best rank-one approximation of a real partially symmetric rectangular tensor is investigated.

Keywords

E-characteristic polynomial / rectangular tensor / E-singular value / rank-one approximation

References

[1] Qi L Q. Eigenvalues of a real supersymmetric tensor[J]. Journal of Symbolic Computation, 2005, 40(6): 1302-1324.

[2] Lim L H. Singular values and eigenvalues of tensors: A variational approach[C]. In: Proceedings of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing. Puerto Vallarta, Mexico, 2005

[3] Qi L Q, Wang Y J, Wu E X. D-eigenvalues of diffusion kurtosis tensors[J]. Journal of Computational and Applied Mathematics, 2008, 221(1): 150-157.

[4] Ng M, Qi L Q, Zhou G L. Finding the largest eigenvalue of a nonnegative tensor[J]. SIAM Journal on Matrix Analysis and Applications, 2009, 31(3): 1090-1099.

[5] Qi L Q, Wang F, Wang Y J. Z-eigenvalue methods for a global polynomial optimization problem[J]. Mathematical Programming, 2009, 118(2): 301-316.

[6] De Lathauwer L, De Moor B, Vandewalle J. On the best rank-1 and rank-(R ₁,R ₂,...,R _N) approximation of higherorder tensors[J]. SIAM Journal on Matrix Analysis and Applications, 2000, 21(4): 1324-1342.

[7] Chang K, Qi L Q, Zhou G L. Singular values of a real rectangular tensor[J]. Journal of Mathematical Analysis and Applications, 2010, 370(1): 284-294.

[8] Zhao N. E-singular values of a real partially symmetric rectangular tensor[J]. Journal of Shandong University( Natural Science), 2012, 47(10): 34-37.

[9] Qi L Q. Eigenvalues and invariants of tensors[J]. Journal of Mathematical Analysis and Applications, 2007, 325(2): 1363-1377.

[10] Li A M, Qi L Q, Zhang B. E-characteristic polynomials of tensors[J]. Communications in Mathematical Sciences, 2013, 11(1): 33-53.

[11] Hu S L, Qi L Q. The E-characteristic polynomial of a tensor of dimension 2[J]. Applied Mathematics Letters, 2013, 26(2): 225-231.

[12] Qi L Q, Dai H H, Han D R. Conditions for strong ellipticity and M-eigenvalues[J]. Frontiers of Mathematics in China, 2009, 4(2): 349-364.

[13] Ling C, Nie J W, Qi L Q, et al. Biquadratic optimization over unit spheres and semidefinite programming relaxations[J]. SIAM Journal on Optimization, 2009, 20(3): 1286-1310.

[14] Dahl G, Leinaas J M, Myrheim J, et al. A tensor product matrix approximation problem in quantum physics[J]. Linear Algebra and Its Applications, 2007, 420(2/3): 711-725.

[15] Wang Y J, Qi L Q, Zhang X Z. A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor[J]. Numerical Linear Algebra with Applications, 2009, 16(7): 589-601.

[16] Cox D A, Little J, O’Shea D. Using Algebraic Geometry[M]. 1998, New York, USA: Springer-Verlag.